YES

Problem:
 app(app(map(),f),nil()) -> nil()
 app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs))
 app(sum(),app(app(cons(),x),xs)) -> app(app(plus(),x),app(sum(),xs))
 app(size(),app(app(node(),x),xs)) -> app(s(),app(sum(),app(app(map(),size()),xs)))
 app(app(plus(),0()),x) -> 0()
 app(app(plus(),app(s(),x)),y) -> app(s(),app(app(plus(),x),y))

Proof:
 Uncurry Processor:
  map2(f,nil()) -> nil()
  map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
  sum1(cons2(x,xs)) -> plus2(x,sum1(xs))
  size1(node2(x,xs)) -> s1(sum1(map2(size(),xs)))
  plus2(0(),x) -> 0()
  plus2(s1(x),y) -> s1(plus2(x,y))
  app(map1(x4),x5) -> map2(x4,x5)
  app(map(),x5) -> map1(x5)
  app(cons1(x4),x5) -> cons2(x4,x5)
  app(cons(),x5) -> cons1(x5)
  app(sum(),x5) -> sum1(x5)
  app(plus1(x4),x5) -> plus2(x4,x5)
  app(plus(),x5) -> plus1(x5)
  app(size(),x5) -> size1(x5)
  app(node1(x4),x5) -> node2(x4,x5)
  app(node(),x5) -> node1(x5)
  app(s(),x5) -> s1(x5)
  DP Processor:
   DPs:
    map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
    map{2,#}(f,cons2(x,xs)) -> app#(f,x)
    sum{1,#}(cons2(x,xs)) -> sum{1,#}(xs)
    sum{1,#}(cons2(x,xs)) -> plus{2,#}(x,sum1(xs))
    size{1,#}(node2(x,xs)) -> map{2,#}(size(),xs)
    size{1,#}(node2(x,xs)) -> sum{1,#}(map2(size(),xs))
    plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
    app#(map1(x4),x5) -> map{2,#}(x4,x5)
    app#(sum(),x5) -> sum{1,#}(x5)
    app#(plus1(x4),x5) -> plus{2,#}(x4,x5)
    app#(size(),x5) -> size{1,#}(x5)
   TRS:
    map2(f,nil()) -> nil()
    map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
    sum1(cons2(x,xs)) -> plus2(x,sum1(xs))
    size1(node2(x,xs)) -> s1(sum1(map2(size(),xs)))
    plus2(0(),x) -> 0()
    plus2(s1(x),y) -> s1(plus2(x,y))
    app(map1(x4),x5) -> map2(x4,x5)
    app(map(),x5) -> map1(x5)
    app(cons1(x4),x5) -> cons2(x4,x5)
    app(cons(),x5) -> cons1(x5)
    app(sum(),x5) -> sum1(x5)
    app(plus1(x4),x5) -> plus2(x4,x5)
    app(plus(),x5) -> plus1(x5)
    app(size(),x5) -> size1(x5)
    app(node1(x4),x5) -> node2(x4,x5)
    app(node(),x5) -> node1(x5)
    app(s(),x5) -> s1(x5)
   TDG Processor:
    DPs:
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     sum{1,#}(cons2(x,xs)) -> sum{1,#}(xs)
     sum{1,#}(cons2(x,xs)) -> plus{2,#}(x,sum1(xs))
     size{1,#}(node2(x,xs)) -> map{2,#}(size(),xs)
     size{1,#}(node2(x,xs)) -> sum{1,#}(map2(size(),xs))
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     app#(map1(x4),x5) -> map{2,#}(x4,x5)
     app#(sum(),x5) -> sum{1,#}(x5)
     app#(plus1(x4),x5) -> plus{2,#}(x4,x5)
     app#(size(),x5) -> size{1,#}(x5)
    TRS:
     map2(f,nil()) -> nil()
     map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
     sum1(cons2(x,xs)) -> plus2(x,sum1(xs))
     size1(node2(x,xs)) -> s1(sum1(map2(size(),xs)))
     plus2(0(),x) -> 0()
     plus2(s1(x),y) -> s1(plus2(x,y))
     app(map1(x4),x5) -> map2(x4,x5)
     app(map(),x5) -> map1(x5)
     app(cons1(x4),x5) -> cons2(x4,x5)
     app(cons(),x5) -> cons1(x5)
     app(sum(),x5) -> sum1(x5)
     app(plus1(x4),x5) -> plus2(x4,x5)
     app(plus(),x5) -> plus1(x5)
     app(size(),x5) -> size1(x5)
     app(node1(x4),x5) -> node2(x4,x5)
     app(node(),x5) -> node1(x5)
     app(s(),x5) -> s1(x5)
    graph:
     size{1,#}(node2(x,xs)) -> sum{1,#}(map2(size(),xs)) ->
     sum{1,#}(cons2(x,xs)) -> plus{2,#}(x,sum1(xs))
     size{1,#}(node2(x,xs)) -> sum{1,#}(map2(size(),xs)) ->
     sum{1,#}(cons2(x,xs)) -> sum{1,#}(xs)
     size{1,#}(node2(x,xs)) -> map{2,#}(size(),xs) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     size{1,#}(node2(x,xs)) -> map{2,#}(size(),xs) ->
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     sum{1,#}(cons2(x,xs)) -> plus{2,#}(x,sum1(xs)) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     sum{1,#}(cons2(x,xs)) -> sum{1,#}(xs) ->
     sum{1,#}(cons2(x,xs)) -> plus{2,#}(x,sum1(xs))
     sum{1,#}(cons2(x,xs)) -> sum{1,#}(xs) ->
     sum{1,#}(cons2(x,xs)) -> sum{1,#}(xs)
     app#(plus1(x4),x5) -> plus{2,#}(x4,x5) ->
     plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     app#(map1(x4),x5) -> map{2,#}(x4,x5) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     app#(map1(x4),x5) -> map{2,#}(x4,x5) ->
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
     app#(size(),x5) -> size{1,#}(x5) ->
     size{1,#}(node2(x,xs)) -> sum{1,#}(map2(size(),xs))
     app#(size(),x5) -> size{1,#}(x5) ->
     size{1,#}(node2(x,xs)) -> map{2,#}(size(),xs)
     app#(sum(),x5) -> sum{1,#}(x5) ->
     sum{1,#}(cons2(x,xs)) -> plus{2,#}(x,sum1(xs))
     app#(sum(),x5) -> sum{1,#}(x5) ->
     sum{1,#}(cons2(x,xs)) -> sum{1,#}(xs)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(size(),x5) -> size{1,#}(x5)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(plus1(x4),x5) -> plus{2,#}(x4,x5)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(sum(),x5) -> sum{1,#}(x5)
     map{2,#}(f,cons2(x,xs)) -> app#(f,x) ->
     app#(map1(x4),x5) -> map{2,#}(x4,x5)
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs) ->
     map{2,#}(f,cons2(x,xs)) -> app#(f,x)
     map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs) -> map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
    SCC Processor:
     #sccs: 3
     #rules: 7
     #arcs: 21/121
     DPs:
      size{1,#}(node2(x,xs)) -> map{2,#}(size(),xs)
      map{2,#}(f,cons2(x,xs)) -> map{2,#}(f,xs)
      map{2,#}(f,cons2(x,xs)) -> app#(f,x)
      app#(map1(x4),x5) -> map{2,#}(x4,x5)
      app#(size(),x5) -> size{1,#}(x5)
     TRS:
      map2(f,nil()) -> nil()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      sum1(cons2(x,xs)) -> plus2(x,sum1(xs))
      size1(node2(x,xs)) -> s1(sum1(map2(size(),xs)))
      plus2(0(),x) -> 0()
      plus2(s1(x),y) -> s1(plus2(x,y))
      app(map1(x4),x5) -> map2(x4,x5)
      app(map(),x5) -> map1(x5)
      app(cons1(x4),x5) -> cons2(x4,x5)
      app(cons(),x5) -> cons1(x5)
      app(sum(),x5) -> sum1(x5)
      app(plus1(x4),x5) -> plus2(x4,x5)
      app(plus(),x5) -> plus1(x5)
      app(size(),x5) -> size1(x5)
      app(node1(x4),x5) -> node2(x4,x5)
      app(node(),x5) -> node1(x5)
      app(s(),x5) -> s1(x5)
     Subterm Criterion Processor:
      simple projection:
       pi(map{2,#}) = 1
       pi(app#) = 1
       pi(size{1,#}) = 0
      problem:
       DPs:
        app#(map1(x4),x5) -> map{2,#}(x4,x5)
        app#(size(),x5) -> size{1,#}(x5)
       TRS:
        map2(f,nil()) -> nil()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        sum1(cons2(x,xs)) -> plus2(x,sum1(xs))
        size1(node2(x,xs)) -> s1(sum1(map2(size(),xs)))
        plus2(0(),x) -> 0()
        plus2(s1(x),y) -> s1(plus2(x,y))
        app(map1(x4),x5) -> map2(x4,x5)
        app(map(),x5) -> map1(x5)
        app(cons1(x4),x5) -> cons2(x4,x5)
        app(cons(),x5) -> cons1(x5)
        app(sum(),x5) -> sum1(x5)
        app(plus1(x4),x5) -> plus2(x4,x5)
        app(plus(),x5) -> plus1(x5)
        app(size(),x5) -> size1(x5)
        app(node1(x4),x5) -> node2(x4,x5)
        app(node(),x5) -> node1(x5)
        app(s(),x5) -> s1(x5)
      SCC Processor:
       #sccs: 0
       #rules: 0
       #arcs: 9/4
       
     
     DPs:
      sum{1,#}(cons2(x,xs)) -> sum{1,#}(xs)
     TRS:
      map2(f,nil()) -> nil()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      sum1(cons2(x,xs)) -> plus2(x,sum1(xs))
      size1(node2(x,xs)) -> s1(sum1(map2(size(),xs)))
      plus2(0(),x) -> 0()
      plus2(s1(x),y) -> s1(plus2(x,y))
      app(map1(x4),x5) -> map2(x4,x5)
      app(map(),x5) -> map1(x5)
      app(cons1(x4),x5) -> cons2(x4,x5)
      app(cons(),x5) -> cons1(x5)
      app(sum(),x5) -> sum1(x5)
      app(plus1(x4),x5) -> plus2(x4,x5)
      app(plus(),x5) -> plus1(x5)
      app(size(),x5) -> size1(x5)
      app(node1(x4),x5) -> node2(x4,x5)
      app(node(),x5) -> node1(x5)
      app(s(),x5) -> s1(x5)
     Subterm Criterion Processor:
      simple projection:
       pi(sum{1,#}) = 0
      problem:
       DPs:
        
       TRS:
        map2(f,nil()) -> nil()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        sum1(cons2(x,xs)) -> plus2(x,sum1(xs))
        size1(node2(x,xs)) -> s1(sum1(map2(size(),xs)))
        plus2(0(),x) -> 0()
        plus2(s1(x),y) -> s1(plus2(x,y))
        app(map1(x4),x5) -> map2(x4,x5)
        app(map(),x5) -> map1(x5)
        app(cons1(x4),x5) -> cons2(x4,x5)
        app(cons(),x5) -> cons1(x5)
        app(sum(),x5) -> sum1(x5)
        app(plus1(x4),x5) -> plus2(x4,x5)
        app(plus(),x5) -> plus1(x5)
        app(size(),x5) -> size1(x5)
        app(node1(x4),x5) -> node2(x4,x5)
        app(node(),x5) -> node1(x5)
        app(s(),x5) -> s1(x5)
      Qed
     
     DPs:
      plus{2,#}(s1(x),y) -> plus{2,#}(x,y)
     TRS:
      map2(f,nil()) -> nil()
      map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
      sum1(cons2(x,xs)) -> plus2(x,sum1(xs))
      size1(node2(x,xs)) -> s1(sum1(map2(size(),xs)))
      plus2(0(),x) -> 0()
      plus2(s1(x),y) -> s1(plus2(x,y))
      app(map1(x4),x5) -> map2(x4,x5)
      app(map(),x5) -> map1(x5)
      app(cons1(x4),x5) -> cons2(x4,x5)
      app(cons(),x5) -> cons1(x5)
      app(sum(),x5) -> sum1(x5)
      app(plus1(x4),x5) -> plus2(x4,x5)
      app(plus(),x5) -> plus1(x5)
      app(size(),x5) -> size1(x5)
      app(node1(x4),x5) -> node2(x4,x5)
      app(node(),x5) -> node1(x5)
      app(s(),x5) -> s1(x5)
     Subterm Criterion Processor:
      simple projection:
       pi(plus{2,#}) = 0
      problem:
       DPs:
        
       TRS:
        map2(f,nil()) -> nil()
        map2(f,cons2(x,xs)) -> cons2(app(f,x),map2(f,xs))
        sum1(cons2(x,xs)) -> plus2(x,sum1(xs))
        size1(node2(x,xs)) -> s1(sum1(map2(size(),xs)))
        plus2(0(),x) -> 0()
        plus2(s1(x),y) -> s1(plus2(x,y))
        app(map1(x4),x5) -> map2(x4,x5)
        app(map(),x5) -> map1(x5)
        app(cons1(x4),x5) -> cons2(x4,x5)
        app(cons(),x5) -> cons1(x5)
        app(sum(),x5) -> sum1(x5)
        app(plus1(x4),x5) -> plus2(x4,x5)
        app(plus(),x5) -> plus1(x5)
        app(size(),x5) -> size1(x5)
        app(node1(x4),x5) -> node2(x4,x5)
        app(node(),x5) -> node1(x5)
        app(s(),x5) -> s1(x5)
      Qed